XIV

PREFACE

to an approach close to the original one of Bender and Aschbacher, for the sake of

efficiency.

Chapter 3 is devoted to five "p-component pre-uniqueness theorems", Theorems

PU1-PU5. The common theme is a maximal subgroup M which has a p-component

K such that the centralizer W of KjOv (K) has p-rank at least 2 and such that

CG(V) M for every non-identity p-element of W. This type of situation will

arise typically when K is maximal in some ordering on the set of p-components of

centralizers of p-elements of G, and the p'-cores

O

P

/(GG(X)) , X

G W, have already

been assembled, for example by the signalizer functor method or by an assumption

that they are trivial. The generic conclusion is that M is a strongly p-embedded

subgroup of G, which yields an immediate contradiction if p = 2. When p is odd

and the DC-proper simple group G has even type, it will later be shown that again G

does not exist. Thus the eventual import of Chapter 3 will be to establish in general

that maximal p-components of centralizers of p-elements have centralizers of p-rank

1. (Of course there are counterexamples to this statement, for example when p = 2

and G is an alternating group.) Historically results of this type were established

first for p = 2 by Powell and Thwaites, whose ideas are incorporated into Section

17 of Chapter 3. Shortly thereafter, Aschbacher proved his Component Theorem, a

more definitive version for p = 2. Robert Gilman gave a somewhat different proof

of Aschbacher's result. Many of the ideas of Aschbacher and Gilman also appear

here, but some of their delicate analysis is replaced by a detailed consideration of

DC-groups. The results proved here are new in the case that p is odd. We remark

that in the proof of the first major pre-uniqueness theorem, Theorem PUi, the

case p = 2 is handled fairly quickly thanks to Aschbacher's criterion for a strongly

embedded subgroup (Theorem ZD). Thus in Sections 7-15 of Chapter 3 the prime

p is odd.

The structure and embedding of p-component uniqueness subgroups when p = 2

and K has 2-rank one is somewhat exceptional. In particular the main assertion of

Theorem PU4, that terminal components are standard, is not valid in this situa-

tion. In other words, K could commute elementwise with a conjugate. Historically

Aschbacher and Richard Foote were able to show that there could be only one

such conjugate, and we prove a similar result in Theorem PU5. Thanks are due to

Professor Foote, who suggested years ago that such a result ought to be simple to

prove.

As noted above, with the exception of Theorem ZD and its corollaries Theorems

SE and SZ, all the results are proved for a DC-proper simple group G. The proof

thus can and does rely heavily on the theory of almost simple DC-groups extablished

in pU]. Those DC-group properties essential for Chapters 2 and 3 are collected

in Chapter 4 of this volume and either extend or follow directly from the theory

presented in our preceding volume [IA] . A much briefer Chapter 1 similarly extends

our second volume [IQ] with some "general" (as opposed to DC-group theoretic)

results pertinent to our task. Notable here is some theory of permutation groups

underlying the proof of Theorem ZD.

In references (even within this volume), we shall specify the four chapters of

this volume as Hi, II2, II3 and II4, respectively.

We are grateful to Michael O'Nan for his assistance with the proof of Theorem

3.2 of Chapter 1; to Sergey Shpectorov for his enthusiastic support and constructive

comments during the preparation of Chapter 2; to Michael Aschbacher and Hel-

mut Bender for their support and helpful comments and suggestions; and to Inna